I think you're missing something. Producing a vaccine is nothing complicated, it is developing the vaccine that takes time so why should he be asking for it to be "only for the US"?
Unless someone want to monopolize the vaccine fir monetary or political reasons, then there is zero reason to try what Trump seems to be trying.
As for media trying to portray Trump as bad, doing so is easy - all that needs doing is to let him speak and then showing how much he is lying. That and his behavior in general clearly shows what kind of person he is, like fx. he claims to be really smart because he has an uncle that worked at M.I.T.!